Borderline Gradient Continuity for the Normalized p-Parabolic Operator

Abstract

AbstractIn this paper, we prove gradient continuity estimates for viscosity solutions to $$\Delta _{p}^N u - u_t= f$$ Δ p N u - u t = f in terms of the scaling critical $$L(n+2,1 )$$ L ( n + 2 , 1 ) norm of f, where $$\Delta _{p}^N$$ Δ p N is the game theoretic normalized $$p-$$ p - Laplacian operator defined in (1.2) below. Our main result, Theorem 2.5 constitutes borderline gradient continuity estimate for u in terms of the modified parabolic Riesz potential $$\textbf{P}^{f}_{n+1}$$ P n + 1 f as defined in (2.9) below. Moreover, for $$f \in L^{m}$$ f ∈ L m with $$m>n+2$$ m > n + 2 , we also obtain Hölder continuity of the spatial gradient of the solution u, see Theorem 2.6 below. This improves the gradient Hölder continuity result in Attouchi and Parviainen (Commun Contemp Math 20(4):1750035, 2018) which considers bounded f. Our main results Theorem 2.5 and Theorem 2.6 are parabolic analogues of those in Banerjee and Munive (Commun Contemp Math 22(8):1950069, 2020). Moreover differently from that in Attouchi and Parviainen (Commun Contemp Math 20(4):1750035, 2018), our approach is independent of the Ishii–Lions method which is crucially used in Attouchi and Parviainen (Commun Contemp Math 20(4):1750035, 2018) to obtain Lipschitz estimates for homogeneous perturbed equations as an intermediate step.